# set theory – Coding the universe into a real over better core models

One of the most incredible results in modern set theory, due to Jensen, is that given any model of $$sf ZFC$$, there is a class forcing which adds a real number $$r$$ and in the extension $$V=L[r]$$. Moreover, if we assume some basic things like $$sf GCH$$, then cardinals are not collapsed, so in particular things like inaccessible cardinals are preserved.

So in Jensen’s result the “coding model” is $$L$$. But $$L[x]$$ is somewhat of a dull model. It doesn’t have any sharps, measurable cardinals, or larger cardinals. Even those with reasonably canonical inner models.

Question. Given any reasonably canonical core model $$K$$, can we code the universe into a real over $$K$$ while preserving large cardinals that are captured by $$K$$?

In other words, can we code the universe into a real while preserving measurable cardinals? Can we code the universe into a real while preserving strong, Woodin, etc.?

And the obvious follow-up question, what is the bare minimum needed from an inner model $$M$$ to be a coding model?